#7623640

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
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

1962 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)(7,15)(8,20)(9,25)(10,13)(11,18)(12,23)(14,22)(16,33)(17,35)(19,34)(21,31)(24,32)(26,29)(27,30)(28,36);(5,7,10,19)(6,11,13,16)(8,21,24,36)(9,26,22,17)(12,29,30,35)(14,31,25,28)(15,32,34,20)(18,27,33,23);(2,4)(5,6)(7,16)(8,12)(9,14)(10,13)(11,19)(15,33)(17,31)(18,34)(20,23)(21,35)(22,25)(24,30)(26,28)(27,32)(29,36);(5,8)(6,12)(7,17)(9,27)(10,24)(11,28)(13,30)(14,32)(15,34)(16,31)(18,33)(19,26)(20,25)(22,23);(6,13)(11,16)(12,30)(14,25)(15,34)(20,32)(28,31)(29,35)

Polycyclic presentation:

2^9|b^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,c^b=ci,d^b=df,f^b=fh,h^b=hi,e^c=ei,f^d=fh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343457, #72343835, #72343839
maximal normal subgroup of	#72343457, #72343835, #72343839

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530836);