#110036

order	512
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1436
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,f^2=h,b^a=bd,c^a=ce,g^a=gh,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17232);