#7591007

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(9,10,12,14)(11,13,15,16)(17,18,23,26)(19,27,31,44)(20,32,34,25)(21,37,30,24)(22,39,38,28)(29,45,33,42)(35,40,47,41)(36,43,46,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30,23,31,38,21)(18,24,28,27,26,37,39,44)(20,33,36,35,34,29,46,47)(25,41,43,45,32,40,48,42);(2,5)(6,7)(17,20,23,34)(18,25,26,32)(19,29,31,33)(21,35,30,47)(22,36,38,46)(24,40,37,41)(27,42,44,45)(28,43,39,48);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,26)(20,34)(24,37)(27,44)(28,39)(29,33)(35,47)(36,46);(17,21,38,31,23,30,22,19)(18,27,39,24,26,44,28,37)(20,35,46,33,34,47,36,29)(25,42,48,40,32,45,43,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

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, 7498203);