#190505

order	512
nilpotency class	3
isoclinism	isoclinic class of #184811
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=ei,g^2=i,b^a=be,c^a=cf,f^a=fh,d^c=di,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97701);