#251905

order	512
nilpotency class	3
isoclinism	isoclinic class of #251883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

952 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283717, #72283912, #72284060, …
maximal normal subgroup of	#72283717, #72283912, #72284060, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159101);