#7504206

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 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)(13,14,18,23,17,22,31,21)(15,24,27,37,26,19,33,35)(16,28,30,36,29,38,32,20)(25,40,42,34,41,44,43,39);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,26)(14,19,22,24)(16,25,29,41)(18,27,31,33)(20,34,36,39)(21,37,23,35)(28,44,38,40)(30,42,32,43);(2,4)(13,16,17,29)(14,20,22,36)(15,25,26,41)(18,32,31,30)(19,34,24,39)(21,28,23,38)(27,43,33,42)(35,40,37,44);(6,11)(7,12)(14,21)(15,26)(18,31)(19,35)(20,28)(22,23)(24,37)(25,41)(30,32)(34,40)(36,38)(39,44);(5,8,9,10)(6,7,11,12)(14,22)(15,26)(20,36)(21,23)(25,41)(27,33)(28,38)(42,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=fi,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, 7411402);