#6372716

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4456 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341707, #72341746
maximal normal subgroup of	#72341707, #72341746

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279912);