#358913

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
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	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266109);