#422900

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1933 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)(7,15)(8,20)(9,24)(10,14)(11,18)(12,22)(13,26)(16,33)(17,27)(19,32)(21,31)(23,30)(25,29)(28,36)(34,35);(1,2,3,4)(5,7,9,18,10,19,26,33)(6,11,13,15,14,16,24,32)(8,21,22,35,23,36,25,17)(12,28,20,34,29,31,30,27);(5,6)(7,16,19,11)(8,12)(9,13)(10,14)(15,18,32,33)(17,34,35,27)(20,22)(21,31,36,28)(23,29)(24,26)(25,30);(2,4)(5,8,10,23)(6,12,14,29)(7,17,19,35)(9,25,26,22)(11,27,16,34)(13,30,24,20)(15,31,32,28)(18,36,33,21)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251308, #72262403, #72262421
maximal normal subgroup of	#72251308, #72262403, #72262421

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330096);