#394264

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341298, #72341344, #72341353
maximal normal subgroup of	#72341298, #72341344, #72341353

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301460);