#7564829

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45196	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=gi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei

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, 7472025);