#204505

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1246 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,g^b=gh

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, 111701);