#7537914

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,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, 7445110);