#6376277

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283473);