#417846

order	512
nilpotency class	3
isoclinism	isoclinic class of #417489
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

2794 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)(7,13)(8,20)(9,12)(10,26)(11,15)(14,24)(16,33)(17,19)(18,30)(21,28)(22,25)(23,32)(27,29)(31,35)(34,36);(5,7)(6,13)(8,21,23,36)(9,17)(10,27)(11,18)(12,19)(14,31,22,33)(15,30)(16,25,35,24)(20,34,32,28)(26,29);(5,8)(6,14)(7,16)(9,22)(10,25)(11,24)(12,23)(13,28)(15,32)(17,34)(18,36)(19,35)(20,26)(21,27)(29,31)(30,33);(2,4)(5,9,12,6)(7,17,19,13)(8,22,23,14)(10,26,11,15)(16,34,35,28)(18,30,27,29)(20,24,32,25)(21,31,36,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313084, #72313151, #72313164
maximal normal subgroup of	#72313084, #72313151, #72313164

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325042);