#7607430

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=gh,e^2=fgi,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, 7514626);