#6545026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#3269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2020 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)(13,14,19,31,18,22,17,23)(15,20,32,42,25,35,26,34)(16,27,28,21,30,37,29,36)(24,38,40,33,39,43,41,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,26)(18,25)(19,32)(21,33)(22,35)(23,34)(27,38)(28,40)(29,41)(30,39)(31,42)(36,44)(37,43);(6,9)(10,12)(13,16,18,30)(14,21,22,36)(15,24,25,39)(17,28,19,29)(20,33,35,44)(23,37,31,27)(26,40,32,41)(34,43,42,38);(5,7)(6,10)(8,11)(9,12)(14,22)(15,25)(21,36)(23,31)(24,39)(26,32)(27,37)(40,41);(7,11)(10,12)(13,17,18,19)(14,23,22,31)(15,26,25,32)(16,28,30,29)(20,34,35,42)(21,37,36,27)(24,40,39,41)(33,43,44,38)

Polycyclic presentation:

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

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, 6452222);