#227696

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×C₄×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

856 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)(5,6,9,7,8,10,12,11)(14,18)(16,24)(19,25)(20,21)(22,28)(26,27);(5,7,9,10,8,11,12,6)(13,14)(15,19)(16,20)(17,21)(18,24)(22,26)(23,27)(25,28);(2,4)(13,15,17,23)(14,19,21,27)(16,22,24,28)(18,25,20,26);(14,20)(16,24)(18,21)(19,26)(22,28)(25,27)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dei,e^b=eh,e^d=eh

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, 134892);