#464082

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1439 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)(5,6,11,16)(7,13,8,20)(9,23,27,32)(10,28,21,15)(12,19,30,18)(14,26,33,25)(17,34,22,29)(24,31,35,36);(5,7,10,18,11,8,21,19)(6,12,15,13,16,30,28,20)(9,17,26,24,27,22,25,35)(14,29,32,31,33,34,23,36);(2,4)(5,8)(6,13)(7,11)(9,24)(10,18)(12,15)(14,29)(16,20)(17,26)(19,21)(22,25)(23,31)(27,35)(28,30)(32,36)(33,34);(5,9)(6,14)(7,17)(8,22)(10,26)(11,27)(12,29)(13,31)(15,32)(16,33)(18,24)(19,35)(20,36)(21,25)(23,28)(30,34)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 371278);