#7595211

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,24,30,25)(20,31,34,41)(21,35,37,48)(22,39,32,27)(26,43,45,38)(29,44,46,47)(33,40,36,42);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,28,25)(20,22,34,32)(21,33,37,36)(26,44,45,47)(27,31,39,41)(29,43,46,38)(35,40,48,42);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,32,19,23,34,22,30)(18,25,39,41,28,24,27,31)(21,36,43,46,37,33,38,29)(26,40,48,47,45,42,35,44);(9,11,12,16)(10,14,13,15)(17,21,32,43,23,37,22,38)(18,26,39,48,28,45,27,35)(19,29,34,36,30,46,20,33)(24,40,31,47,25,42,41,44);(2,5)(6,8)(21,37)(26,45)(29,46)(33,36)(35,48)(38,43)(40,42)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=gi,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,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, 7502407);