#7200359

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2282 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)(11,13)(12,14)(15,16)(17,18)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,34)(28,42)(29,40)(33,39)(36,37)(38,48)(41,43)(44,45)(46,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(20,32)(24,37)(26,35)(27,40)(28,43)(33,47)(44,48);(11,15)(13,16)(17,19,21,29)(18,24,23,37)(20,28,32,43)(22,36,34,30)(25,38,31,45)(26,40,35,27)(33,48,47,44)(39,42,46,41);(3,8)(5,7)(10,14)(11,15)(17,20)(18,25)(19,28)(21,32)(22,33)(23,31)(24,38)(26,39)(27,41)(29,43)(30,44)(34,47)(35,46)(36,48)(37,45)(40,42);(2,6)(5,7)(11,15)(13,16)(18,23)(20,32)(24,37)(26,35)(27,40)(28,43)(33,47)(44,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107555);