#455688

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 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)(7,11)(8,13)(9,14)(10,12)(15,16)(17,25)(18,24)(19,29)(20,28)(21,26)(22,23)(27,30)(31,35)(32,36)(33,34);(5,7,16,19,9,17,10,22)(6,11,12,23,14,25,15,29)(8,18,30,32,20,31,21,33)(13,24,26,34,28,35,27,36);(6,12)(7,17)(8,20)(10,16)(11,23)(13,27)(14,15)(18,31)(24,34)(25,29)(26,28)(35,36);(2,4)(5,8)(6,13)(7,18)(9,20)(10,21)(11,24)(12,26)(14,28)(15,27)(16,30)(17,31)(19,32)(22,33)(23,34)(25,35)(29,36)

Polycyclic presentation:

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

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, 362884);