#121615

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,g^b=gh,g^c=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28811);