#322872

order	512
nilpotency class	3
isoclinism	isoclinic class of #322813
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#30258	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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)(3,10)(4,6)(5,9)(7,12)(8,11)(13,14)(15,16)(17,18,23,32,21,26,22,27)(19,20,29,31,25,24,28,30);(2,6,9,8)(7,14,15,10)(17,19,21,25)(18,24,26,20)(22,28,23,29)(27,31,32,30);(1,2)(3,7)(4,8)(5,9)(6,11)(10,12)(13,15)(14,16)(17,20,21,24)(18,25,26,19)(22,31,23,30)(27,28,32,29);(1,3,11,16,5,13,4,12)(2,7,6,14,9,15,8,10)(18,26)(19,25)(27,32)(28,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230068);