#7409989

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=i,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7317185);