#226201

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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×C₄⋊C₄
prime radical	C₂×#135	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 133397);