#215696

order	512
nilpotency class	3
isoclinism	isoclinic class of #162550
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

840 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,c^2=e,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fi,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122892);