#213885

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1038 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,27,30,23)(20,24)(21,33,34,25)(22,26)(28,35,42,37)(29,36)(31,44,45,38)(32,39)(40,43)(41,47)(46,48);(4,12)(7,14)(10,16)(13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,37)(25,36)(26,27)(31,41)(32,42)(33,40)(34,43)(35,39)(38,47)(44,46)(45,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,22,32)(18,24,26,39)(19,28,30,42)(21,31,34,45)(23,35,27,37)(25,38,33,44)(29,41,43,48)(36,46,40,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,22,34)(18,25,26,33)(19,29,30,43)(20,31,32,45)(23,36,27,40)(24,38,39,44)(28,41,42,48)(35,46,37,47)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340924, #72340928, #72340951, …
maximal normal subgroup of	#72340924, #72340928, #72340951, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121081);