#7186198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#31850	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=fgi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308849, #72308949
maximal normal subgroup of	#72308849, #72308949

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7093394);