#213887

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 956 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)(3,11,13,7)(4,9,5,8)(6,10)(12,16)(14,15)(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);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,16,9,15)(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);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(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);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,16,11,15)(17,21)(18,25)(19,29)(20,31)(22,34)(23,36)(24,38)(26,33)(27,40)(28,41)(30,43)(32,45)(35,46)(37,47)(39,44)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,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	#72268106
maximal normal subgroup of	#72268106
maximal quotient of	#72268110

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121083);