#221126

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
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	#61434	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128322);