#6465726

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,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, 6372922);