#370459

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=efg,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,d^c=di

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, 277655);