#358738

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,g^c=gi

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, 265934);