#7504197

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	#31060	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)(13,14,17,21,18,22,30,36)(15,19,25,32,24,33,40,44)(16,26,27,35,28,34,29,20)(23,37,39,43,38,42,41,31);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,31)(21,32)(22,33)(26,37)(27,39)(28,38)(29,41)(30,40)(34,42)(35,43)(36,44);(2,4)(13,16,18,28)(14,20,22,35)(15,23,24,38)(17,29,30,27)(19,31,33,43)(21,34,36,26)(25,41,40,39)(32,42,44,37);(6,11)(7,12)(14,21)(17,30)(19,32)(20,34)(22,36)(25,40)(26,35)(27,29)(31,42)(33,44)(37,43)(39,41);(5,8,9,10)(6,7,11,12)(14,22)(15,24)(20,35)(21,36)(23,38)(25,40)(26,34)(39,41)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,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, 7411393);