#454400

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
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

1990 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)(7,16)(8,21)(9,25)(10,14)(11,15)(12,17)(13,22)(18,32)(19,26)(20,28)(23,29)(24,35)(27,33)(30,31)(34,36);(5,7,9,17,11,19,10,20)(6,12,14,16,15,28,25,26)(8,18,22,31,24,33,23,34)(13,27,21,30,29,32,35,36);(7,17)(8,22)(9,10)(12,26)(13,21)(14,25)(16,28)(18,33)(19,20)(23,24)(29,35)(30,36);(2,4)(5,8,10,23,11,24,9,22)(6,13,25,35,15,29,14,21)(7,18,20,34,19,33,17,31)(12,27,26,36,28,32,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255603, #72262411, #72262421, …
maximal normal subgroup of	#72255603, #72262411, #72262421, …
maximal quotient of	#72314549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361596);