#7185312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092508);