#182026

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1380 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=i,g^2=i,b^a=be,g^a=gh,e^b=eh,d^c=di

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89222);