#7328914

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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

1319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,31)(27,35,38,50)(28,33,40,45)(30,43,36,34)(32,46,44,39)(37,51,41,54)(42,56,52,49)(47,55,48,53);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,31)(28,40)(32,44)(34,43)(35,50)(37,41)(42,52)(53,55);(1,3)(2,5)(4,8)(6,7)(25,27,36,46,29,38,30,39)(26,32,43,50,31,44,34,35)(28,41,52,53,40,37,42,55)(33,48,56,51,45,47,49,54);(2,6)(3,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,37)(29,40)(30,42)(31,45)(32,47)(34,49)(35,51)(36,52)(38,41)(39,53)(43,56)(44,48)(46,55)(50,54);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(26,31)(27,38)(28,40)(34,43)(39,46)(42,52)(47,48)(51,54)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7236110);