#7622537

order	512
nilpotency class	4
isoclinism	isoclinic class of #62636
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1910 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)(6,11)(8,12)(9,19)(13,21)(14,22)(15,29)(17,25)(18,31)(20,26)(23,24)(27,32)(28,36)(30,35)(33,34);(5,6,12,24,9,15,20,22,10,14,26,29,19,23,8,11)(7,13,25,34,18,28,30,32,16,27,35,36,31,33,17,21);(5,7)(6,13)(8,17)(9,18)(10,16)(11,21)(12,25)(14,27)(15,28)(19,31)(20,30)(22,32)(23,33)(24,34)(26,35)(29,36);(7,16)(13,27)(17,30)(18,31)(21,32)(25,35)(28,33)(34,36);(6,14)(11,22)(13,27)(15,23)(21,32)(24,29)(28,33)(34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,f^2=hi,h^2=i,b^a=bf,f^a=fh,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321030, #72321058
maximal normal subgroup of	#72321030, #72321058

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529733);