#6376508

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2431	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,d^b=di,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283704);