#411850

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

2146 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)(5,6,11,15)(7,16,14,10)(8,12,21,28)(9,23,25,18)(13,29,30,24)(17,20,27,22)(19,32,34,31)(26,33,36,35);(5,7,10,6,11,14,16,15)(8,17,20,28,21,27,22,12)(9,18,24,13,25,23,29,30)(19,31,33,36,34,32,35,26);(2,4)(5,8)(6,12)(7,17)(9,19)(10,22)(11,21)(13,26)(14,27)(15,28)(16,20)(18,31)(23,32)(24,35)(25,34)(29,33)(30,36);(5,9,16,29,11,25,10,24)(6,13,7,18,15,30,14,23)(8,19,20,33,21,34,22,35)(12,26,17,31,28,36,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=hi,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	#72251062, #72251079, #72251089
maximal normal subgroup of	#72251062, #72251079, #72251089
maximal quotient of	#72314549, #72314551

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319046);