#417858

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2753 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)(7,19)(8,24)(9,15)(10,17)(11,16)(12,18)(13,22)(14,25)(20,29)(21,35)(23,34)(26,31)(27,32)(28,30)(33,36);(5,7,10,22,12,23,11,28)(6,13,16,19,18,30,17,34)(8,20,25,21,9,26,27,36)(14,31,24,35,32,29,15,33);(5,8,12,9)(6,14,18,32)(7,20,23,26)(10,25,11,27)(13,29,30,31)(15,16,24,17)(19,33,34,35)(21,28,36,22);(2,4)(5,9,12,8)(6,15,18,24)(7,21,23,36)(10,25,11,27)(13,29,30,31)(14,16,32,17)(19,35,34,33)(20,28,26,22)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252185, #72256284, #72256289
maximal normal subgroup of	#72252185, #72256284, #72256289

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325054);