#6810582

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2029 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717778);