#373427

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1021 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)(5,6)(7,11)(8,10)(9,12)(14,18,20,23)(15,21,22,24);(2,4)(5,7,9,8)(6,10,12,11)(13,14,16,23,17,20,25,18)(15,19,24,27,22,28,21,26);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,15,17,22)(14,19,20,28)(16,24,25,21)(18,26,23,27);(5,8,9,7)(6,11,12,10)(13,15,17,22)(14,19,20,28)(16,24,25,21)(18,26,23,27)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=efg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

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, 280623);