#456647

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1409 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=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, 363843);