#391858

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1750 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,d^2=e,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gi,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, 299054);