#7533734

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2021 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)(7,11)(8,10)(9,12)(13,14,19,24)(15,25,17,32)(16,21,29,36)(18,34,26,23)(20,33,22,28)(27,39,30,42)(31,44,40,38)(35,43,37,41);(1,2)(3,4)(7,12)(9,11)(13,15,26,33,19,17,18,28)(14,20,34,32,24,22,23,25)(16,27,40,43,29,30,31,41)(21,35,44,42,36,37,38,39);(5,7,8,12)(6,9,10,11)(13,16,19,29)(14,21,24,36)(15,27,17,30)(18,31,26,40)(20,35,22,37)(23,38,34,44)(25,39,32,42)(28,41,33,43);(2,4)(6,10)(9,11)(15,17)(16,29)(20,22)(21,36)(25,32)(28,33)(31,40)(38,44);(13,17,18,33,19,15,26,28)(14,22,23,32,24,20,34,25)(16,30,31,43,29,27,40,41)(21,37,38,42,36,35,44,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=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, 7440930);