#7623585

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6)(7,16)(8,21)(9,14)(10,23)(11,24)(12,15)(13,19)(17,36)(18,34)(20,35)(22,29)(25,26)(27,31)(28,30)(32,33);(5,7)(6,13)(8,22)(9,18)(10,27)(11,29)(12,20)(14,31)(15,32)(16,23)(17,24)(19,21)(25,34)(26,33)(28,36)(30,35);(2,4);(5,8)(6,10)(7,17)(9,23)(11,30)(12,25)(13,18)(14,26)(15,28)(16,33)(19,35)(20,31)(21,24)(22,27)(29,32)(34,36);(5,9,12,11)(6,14,15,24)(7,18,20,29)(8,23,25,30)(10,26,28,21)(13,31,32,17)(16,34,35,22)(19,27,33,36)

Polycyclic presentation:

2^9|e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,h^a=hi,d^b=df,f^b=fh,h^b=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#3414, #72321618
maximal normal subgroup of	D₄×#3414, #72321618

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530781);