#7504180

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#31059	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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)(7,8)(9,11)(10,12)(13,14,19,21,17,23,18,24)(15,25,34,37,27,20,28,35)(16,29,33,22,31,40,32,39)(26,41,43,38,30,44,42,36);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,30)(17,27)(18,28)(19,34)(21,35)(22,38)(23,25)(24,37)(26,31)(29,41)(32,43)(33,42)(36,39)(40,44);(2,4)(14,21)(18,19)(20,35)(22,29)(23,24)(25,37)(28,34)(32,33)(36,44)(38,41)(39,40)(42,43);(6,11)(7,12)(13,16,17,31)(14,22,23,39)(15,26,27,30)(18,33,19,32)(20,36,25,38)(21,29,24,40)(28,43,34,42)(35,44,37,41);(5,8,9,10)(6,7,11,12)(15,27)(20,25)(26,30)(28,34)(35,37)(36,38)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411376);