#182367

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,b^a=be,e^a=eh,g^a=gh,c^b=ch,d^b=dhi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322122, #72322134, #72322190
maximal normal subgroup of	#72322122, #72322134, #72322190

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89563);