#181225

order	512
nilpotency class	3
isoclinism	isoclinic class of #181129
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1590 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)(4,7)(6,8)(9,10)(11,19,21,15)(12,16)(13,18)(14,17)(20,31,33,26)(22,32,35,27)(23,28)(24,29)(25,30)(34,39,40,37)(36,38);(2,5,7,6)(9,11,13,22,14,21,25,35)(10,15,18,27,17,19,30,32)(12,20,24,34,23,33,36,40)(16,26,29,37,28,31,38,39);(9,12)(10,16)(11,20)(13,24)(14,23)(15,26)(17,28)(18,29)(19,31)(21,33)(22,34)(25,36)(27,37)(30,38)(32,39)(35,40);(12,23)(16,28)(20,33)(24,36)(26,31)(29,38)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,g^2=i,b^a=be,e^a=eh,g^a=ghi,f^b=fi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318000, #72318393, #72318834, …
maximal normal subgroup of	#72318000, #72318393, #72318834, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88421);