#7562513

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#25787	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2179 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)(8,9)(10,12)(13,14)(15,25)(16,30)(17,20)(18,24)(19,23)(21,29)(22,33)(26,34)(27,41)(28,37)(31,35)(32,38)(36,42)(39,43)(40,44);(1,2)(3,4)(6,9)(10,11)(14,20)(15,26)(16,31)(18,19)(21,25)(22,30)(27,42)(28,29)(32,33)(34,37)(35,38)(39,44);(13,15,19,26,17,28,18,29)(14,21,24,37,20,34,23,25)(16,27,31,40,32,42,33,43)(22,36,38,44,35,41,30,39);(2,4)(13,16,18,33,17,32,19,31)(14,22,23,30,20,35,24,38)(15,27,29,43,28,42,26,40)(21,36,25,39,34,41,37,44);(5,7,8,12)(6,10,9,11)(14,20)(16,32)(21,34)(23,24)(25,37)(27,42)(31,33)(40,43)

Polycyclic presentation:

2^9|c^2=gi,d^2=g,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469709);