#7610955

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

1442 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,33)(21,27)(22,28)(23,29)(25,32)(26,36)(30,41)(31,40)(34,48)(35,45)(37,42)(38,44)(39,47)(43,46);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,32)(18,24,28,25)(20,34,36,46)(21,31,23,37)(26,43,33,48)(27,40,29,42)(30,38,47,35)(39,45,41,44);(10,14)(11,15)(17,19)(18,25)(20,30)(21,37)(22,32)(23,31)(24,28)(26,41)(27,40)(29,42)(33,39)(34,35)(36,47)(38,46)(43,45)(44,48);(10,14)(13,16)(17,20,21,38,22,36,23,35)(18,26,27,45,28,33,29,44)(19,30,37,46,32,47,31,34)(24,39,42,48,25,41,40,43);(1,3,4,8)(2,6,5,7)(17,21,22,23)(18,27,28,29)(19,31,32,37)(20,35,36,38)(24,40,25,42)(26,44,33,45)(30,46,47,34)(39,48,41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7518151);