#7512799

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500661
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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fi,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313303, #72313612, #72313909
maximal normal subgroup of	#72313303, #72313612, #72313909

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7419995);