#360037

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267233);