#6483174

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 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)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,22)(15,23,26,39)(16,20,29,36)(17,31,24,21)(19,32,34,27)(25,38,28,42)(30,44,40,37)(33,43,35,41);(2,4)(7,8)(10,12);(5,7,9,8)(6,10,11,12)(13,15,24,32,18,26,17,27)(14,19,31,39,22,34,21,23)(16,28,40,41,29,25,30,43)(20,35,44,38,36,33,37,42);(5,8,9,7)(6,10,11,12)(13,16,18,29)(14,20,22,36)(15,25,26,28)(17,30,24,40)(19,33,34,35)(21,37,31,44)(23,38,39,42)(27,41,32,43);(6,11)(10,12)(15,26)(16,29)(19,34)(20,36)(23,39)(27,32)(30,40)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390370);