#6472994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2128 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)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,25,34)(16,27,30,41)(17,22,18,32)(21,28,37,29)(24,38,39,44)(26,35,31,36)(33,42,43,40);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,24)(17,31)(18,26)(19,25)(21,33)(22,36)(23,34)(27,38)(28,42)(29,40)(30,39)(32,35)(37,43)(41,44);(7,11)(9,12)(13,16,17,28,19,30,18,29)(14,21,32,41,23,37,22,27)(15,24,31,42,25,39,26,40)(20,33,35,44,34,43,36,38);(7,11)(9,12)(15,25)(20,34)(24,39)(26,31)(33,43)(35,36)(38,44)(40,42);(6,10)(9,12)(13,17,19,18)(14,22,23,32)(15,26,25,31)(16,28,30,29)(20,35,34,36)(21,27,37,41)(24,40,39,42)(33,44,43,38)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=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, 6380190);