#7202569

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18,22,28)(19,30,34,24)(20,26,37,45)(21,29,23,27)(25,38,43,36)(31,40,32,44)(33,41,35,42)(39,47,46,48);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,31)(21,33)(22,34)(23,35)(25,39)(26,44)(27,41)(28,30)(29,42)(32,37)(36,47)(38,48)(40,45)(43,46);(3,7)(5,8)(10,11)(14,15)(17,20)(18,25)(19,31)(21,36)(22,37)(23,38)(24,39)(26,27)(28,43)(29,45)(30,46)(32,34)(33,47)(35,48)(40,42)(41,44);(2,6)(3,7)(9,12,13,16)(10,14,11,15)(17,20)(18,26)(19,32)(21,38)(22,37)(23,36)(24,40)(25,27)(28,45)(29,43)(30,44)(31,34)(33,47)(35,48)(39,42)(41,46);(3,7)(5,8)(10,11)(14,15)(17,21,22,23)(18,27,28,29)(19,33,34,35)(20,36,37,38)(24,41,30,42)(25,26,43,45)(31,47,32,48)(39,44,46,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,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, 7109765);