#6370313

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277509);