#456648

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	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1431 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^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, 363844);