#6628974

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
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4661 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)(5,6,10,14)(7,11,16,25)(8,18,20,28)(9,21,23,13)(12,22,27,19)(15,29,30,36)(17,33,32,26)(24,34,35,31);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,26)(14,25)(18,29)(19,31)(20,30)(21,33)(22,34)(23,32)(27,35)(28,36);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,28)(14,27)(16,30)(17,34)(18,21)(19,23)(25,35)(26,36)(29,33)(31,32);(9,23)(13,21)(17,32)(18,28)(19,22)(26,33)(29,36)(31,34);(7,16)(11,25)(15,30)(17,32)(24,35)(26,33)(29,36)(31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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	#72318225, #72318235, #72318324, …
maximal normal subgroup of	#72318225, #72318235, #72318324, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536170);