#6628975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341700, #72341707, #72341760
maximal normal subgroup of	#72341700, #72341707, #72341760

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536171);