#353820

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261016);