#229145

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×C₄×D₄
prime radical	C₂×#466	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

760 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)(5,6,9,12,8,11,7,10)(14,19,21,26)(20,23,27,28);(5,6,7,10,8,11,9,12)(13,14,15,19,18,21,17,26)(16,23,22,27,25,28,24,20);(2,4)(13,15,18,17)(14,19,21,26)(16,22,25,24)(20,23,27,28);(5,7,8,9)(6,10,11,12)(13,16,17,24,18,25,15,22)(14,20,19,23,21,27,26,28)

Polycyclic presentation:

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

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, 136341);