#7537958

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1835 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)(15,25)(16,28)(17,20)(18,34)(19,22)(21,30)(23,27)(24,35)(26,39)(29,41)(31,38)(32,40)(33,37)(36,44)(42,43);(1,2)(3,4)(7,12)(9,11)(13,15,18,17,19,27,35,33)(14,20,24,25,22,37,34,23)(16,29,32,44,30,26,42,31)(21,38,40,39,28,36,43,41);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,26)(17,31)(18,32)(19,30)(20,36)(22,28)(23,39)(24,40)(25,41)(27,29)(33,44)(34,43)(35,42)(37,38);(2,4)(6,10)(9,11)(14,22)(15,27)(16,30)(17,33)(24,34)(32,42)(36,38)(39,41);(13,17,18,27,19,33,35,15)(14,23,24,20,22,25,34,37)(16,31,32,29,30,44,42,26)(21,39,40,36,28,41,43,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,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, 7445154);