#6377159

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2404	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=cf,e^c=ei,e^d=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, 6284355);