#182288

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,e^2=h,g^2=i,b^a=be,e^a=eh,g^a=gh,c^b=ch,d^c=dhi

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, 89484);