#7622499

order	512
nilpotency class	4
isoclinism	isoclinic class of #62636
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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	C₂×#3584	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3958 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)(6,12)(8,10)(9,21)(13,25)(14,29)(15,26)(17,19)(18,32)(20,23)(22,24)(27,35)(28,36)(30,31)(33,34);(5,6)(7,13)(8,14)(9,22)(10,24)(11,15)(12,21)(16,27)(17,28)(18,33)(19,34)(20,26)(23,29)(25,32)(30,36)(31,35);(5,7)(6,13)(8,17)(9,18)(10,19)(11,16)(12,25)(14,28)(15,27)(20,31)(21,32)(22,33)(23,30)(24,34)(26,35)(29,36);(7,16)(13,27)(17,19)(18,30)(25,35)(28,34)(31,32)(33,36);(5,8,11,10)(6,14,15,24)(7,17,16,19)(9,20,23,21)(12,22,26,29)(13,28,27,34)(18,31,30,32)(25,33,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321086
maximal normal subgroup of	#72321086

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529695);