#455696

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,e^2=i,g^2=i,h^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,g^c=gi,e^d=ei,f^d=fi,h^d=hi

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, 362892);