#7607005

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,23,26)(19,27,30,45)(20,32,34,25)(21,37,31,24)(22,40,39,28)(29,43,33,46)(35,42,38,41)(36,44,47,48);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,31,23,30,39,21)(18,24,28,27,26,37,40,45)(20,33,36,38,34,29,47,35)(25,42,44,43,32,41,48,46);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,23,34)(18,25,26,32)(19,29,30,33)(21,38,31,35)(22,36,39,47)(24,41,37,42)(27,46,45,43)(28,44,40,48);(2,6)(5,7)(11,16)(14,15)(18,26)(19,30)(20,34)(21,31)(28,40)(36,47)(41,42)(43,46);(2,6)(5,7)(17,21,39,30,23,31,22,19)(18,27,40,24,26,45,28,37)(20,35,47,29,34,38,36,33)(25,43,48,42,32,46,44,41)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,e^2=fg,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,e^c=ei,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, 7514201);