#358793

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2778 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=ei

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, 265989);