#455148

order	512
nilpotency class	3
isoclinism	isoclinic class of #454039
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319954, #72320000, #72320049
maximal normal subgroup of	#72319954, #72320000, #72320049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362344);