#7090513

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#44936	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316766, #72316922, #72316943, …
maximal normal subgroup of	#72316766, #72316922, #72316943, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997709);