#204151

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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	#34977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1354 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)(5,6,8,10,9,12,7,11)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(6,10,12,11)(13,15,16,23,18,25,17,24)(14,19,26,28,21,27,20,22);(1,2,3,4)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,28)(25,27);(5,7,9,8)(6,11,12,10)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,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, 111347);