#7308891

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fi,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, 7216087);