#7504190

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	#31046	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 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)(13,14,19,22,17,23,18,24)(15,25,33,35,27,20,28,37)(16,29,32,21,30,38,31,36)(26,40,43,39,41,44,42,34);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,27)(14,20,23,25)(16,26,30,41)(18,28,19,33)(21,34,36,39)(22,37,24,35)(29,44,38,40)(31,42,32,43);(2,4)(13,16,17,30)(14,21,23,36)(15,26,27,41)(18,32,19,31)(20,34,25,39)(22,29,24,38)(28,43,33,42)(35,40,37,44);(6,11)(7,12)(14,22)(15,27)(18,19)(20,35)(21,29)(23,24)(25,37)(26,41)(31,32)(34,40)(36,38)(39,44);(5,8,9,10)(6,7,11,12)(15,27)(20,25)(26,41)(28,33)(34,39)(35,37)(40,44)(42,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^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, 7411386);