#380916

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,d^2=f,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288112);