#7623650

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1982 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)(7,12)(8,19)(9,17)(10,28)(11,14)(13,21)(15,26)(16,25)(18,32)(20,36)(22,34)(23,30)(24,29)(27,35)(31,33);(5,7)(6,8)(9,23)(10,29)(11,18)(12,17)(13,27)(14,22)(15,24)(16,30)(19,26)(20,25)(21,28)(31,34)(32,35)(33,36);(2,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,31)(14,32)(15,30)(16,24)(21,33)(22,35)(23,26)(25,29)(27,34)(28,36);(5,8,11,22)(6,13,14,23)(7,17,18,35)(9,24,27,36)(10,30,25,21)(12,31,32,26)(15,29,33,20)(16,34,28,19);(6,14)(7,18)(13,23)(15,33)(16,28)(17,35)(19,34)(20,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315336, #72315814, #72315876
maximal normal subgroup of	#72315336, #72315814, #72315876

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530846);