#322875

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

682 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)(3,8)(4,12)(5,7)(6,10)(9,15)(11,14)(13,16)(17,18,23,32,21,26,22,27)(19,20,29,31,25,24,28,30);(2,7,10,9)(3,11,13,12)(17,19,21,25)(18,24,26,20)(22,28,23,29)(27,31,32,30);(1,3,6,13)(2,8,10,16)(4,9,14,7)(5,12,15,11)(17,20,21,24)(18,25,26,19)(22,31,23,30)(27,28,32,29);(1,4,5,8,6,14,15,16)(2,3,9,12,10,13,7,11)(18,26)(19,25)(27,32)(28,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=hi,d^2=ei,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, 230071);