#213723

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,g^d=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318422
maximal normal subgroup of	#72318422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120919);