#6394733

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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	#36652	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,22,18,21,28,34)(15,19,25,31,24,30,37,41)(16,20,26,33,27,32,38,42)(23,29,36,40,35,39,43,44);(5,7)(6,9)(8,12)(10,11)(13,15,17,25,18,24,28,37)(14,19,22,31,21,30,34,41)(16,23,26,36,27,35,38,43)(20,29,33,40,32,39,42,44);(2,4)(13,16,17,26,18,27,28,38)(14,20,22,33,21,32,34,42)(15,23,25,36,24,35,37,43)(19,29,31,40,30,39,41,44);(7,12)(9,11)(14,21)(15,24)(20,32)(22,34)(23,35)(25,37)(33,42)(36,43);(6,10)(9,11)(15,24)(19,30)(23,35)(25,37)(29,39)(31,41)(36,43)(40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=dfi,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301929);