#7527608

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1978 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)(7,9,11,12)(13,14)(15,25)(16,28)(17,20)(18,35)(19,22)(21,30)(23,27)(24,36)(26,42)(29,40)(31,37)(32,41)(33,39)(34,38)(43,44);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,17,19,27,36,34)(14,20,24,25,22,38,35,23)(16,29,32,31,30,26,43,33)(21,39,41,42,28,37,44,40);(6,10)(7,11)(13,16)(14,21)(15,26)(17,33)(18,32)(19,30)(20,37)(22,28)(23,42)(24,41)(25,40)(27,29)(31,34)(35,44)(36,43)(38,39);(2,4)(6,10)(9,12)(14,22)(15,27)(16,30)(17,34)(24,35)(32,43)(37,39)(40,42);(13,17,18,27,19,34,36,15)(14,23,24,20,22,25,35,38)(16,31,32,26,30,33,43,29)(21,40,41,39,28,42,44,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,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^c=ei,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, 7434804);