#7062170

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6969366);