#180870

order	512
nilpotency class	2
isoclinism	isoclinic class of #180869
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=i,b^a=be,d^a=di,g^a=gh,c^b=ci,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235211, #72239223, #72239301, …
maximal normal subgroup of	#72235211, #72239223, #72239301, …
maximal quotient of	#72237202

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88066);