#6577017

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2676 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484213);