#189457

order	512
nilpotency class	3
isoclinism	isoclinic class of #184811
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1494 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)(5,6)(7,11,12,9)(8,10)(13,14,17,20,16,18,23,25)(15,19,22,26,21,24,27,28);(1,2,3,4)(14,18)(19,24)(20,25)(26,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,21)(17,22)(18,24)(20,26)(23,27)(25,28);(15,21)(19,24)(22,27)(26,28)

Polycyclic presentation:

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

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, 96653);