#432166

order	512
nilpotency class	3
isoclinism	isoclinic class of #419445
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 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)(5,6,9,12)(7,11)(8,13,19,26)(10,22,21,14)(15,23)(16,24)(17,25)(18,29)(20,34,33,28)(27,32)(30,35)(31,36);(5,7,9,17)(6,11)(8,16)(10,15,21,18)(12,25)(13,27,26,24)(14,29)(19,32)(20,30)(22,23)(28,35,34,36)(31,33);(6,12)(7,15,17,18)(11,23,25,29)(13,26)(14,22)(16,30,32,31)(24,35,27,36)(28,34);(2,4)(5,8,9,19)(6,13,12,26)(7,16)(10,20,21,33)(11,24)(14,28,22,34)(15,30)(17,32)(18,31)(23,35)(25,27)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339362);