#222235

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#3171	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1022 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)(7,8)(9,11)(10,12)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(6,11)(7,9)(8,12)(15,21)(18,24)(22,27)(25,28);(2,4)(5,7,10,9)(6,8,12,11)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(5,8,10,11)(6,7,12,9)(14,19)(15,21)(20,26)(22,27)

Polycyclic presentation:

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

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, 129431);