#457231

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1361 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364427);