#188493

order	512
nilpotency class	3
isoclinism	isoclinic class of #160780
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,f^c=fi,g^c=ghi

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, 95689);