#7409349

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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

1255 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:

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

Polycyclic presentation:

2^9|a^2=hi,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=eh,d^c=dh,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, 7316545);