#6465571

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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
prime radical	C₂×#3150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6372767);