#6465568

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×#3153	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3160 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=hi,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, 6372764);