#6373504

order	512
nilpotency class	3
isoclinism	isoclinic class of #60454
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2090 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=di,e^b=ei,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, 6280700);