#219861

order	512
nilpotency class	3
isoclinism	isoclinic class of #164127
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1247 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127057);