#462212

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

733 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)(5,6,11,15)(7,16)(8,19)(9,13,25,30)(10,14,21,20)(12,27)(17,31)(18,26)(22,32)(23,35)(24,29,34,33)(28,36);(5,7,11,19)(6,12)(8,20)(9,17,25,32)(10,18,21,27)(13,23)(14,16)(15,26)(22,33)(24,28,34,35)(29,31)(30,36);(5,8,21,12,11,16,10,26)(6,7,14,27,15,19,20,18)(9,23,34,31,25,36,24,22)(13,28,29,17,30,35,33,32);(2,4)(5,9,10,24,11,25,21,34)(6,13,14,29,15,30,20,33)(7,17,18,28,19,32,27,35)(8,22,12,23,16,31,26,36)

Polycyclic presentation:

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

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, 369408);