#190367

order	512
nilpotency class	3
isoclinism	isoclinic class of #190218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^b=gh,f^c=fi,g^c=gh

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, 97563);