#353585

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1764 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260781);