#411848

order	512
nilpotency class	3
isoclinism	isoclinic class of #398436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2074 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251058, #72251079, #72251087
maximal normal subgroup of	#72251058, #72251079, #72251087
maximal quotient of	#72314512, #72314513

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319044);