#221135

order	512
nilpotency class	3
isoclinism	isoclinic class of #19883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#61431	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

886 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128331);