#7330770

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1406 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7237966);