#7531941

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1763 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)(12,14)(13,16)(17,18,23,22)(19,24,29,36)(20,25,31,32)(21,26,35,37)(27,38,43,47)(28,39,45,33)(30,40,34,41)(42,48,44,46);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,24,22,36)(20,27,31,43)(21,33,35,39)(25,38,32,47)(26,28,37,45)(30,46,34,48)(40,42,41,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,27)(21,34)(22,32)(23,31)(24,38)(26,41)(28,44)(29,43)(30,35)(33,48)(36,47)(37,40)(39,46)(42,45);(10,14)(13,15)(17,21,23,35)(18,26,22,37)(19,28,29,45)(20,30,31,34)(24,39,36,33)(25,40,32,41)(27,42,43,44)(38,48,47,46);(2,5)(6,8)(10,14)(13,15)(20,31)(25,32)(27,43)(30,34)(38,47)(40,41)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7439137);