#455964

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1130 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)(7,16,20,31)(8,22,24,12)(9,25,27,14)(11,21,30,19)(13,26,33,23)(17,29,18,34)(28,35,32,36);(1,2)(3,4)(5,7,9,19,10,20,27,21)(6,11,14,16,15,30,25,31)(8,18,26,35,24,17,23,36)(12,32,33,34,22,28,13,29);(2,4)(5,8,9,23,10,24,27,26)(6,12,14,13,15,22,25,33)(7,17,19,35,20,18,21,36)(11,28,16,34,30,32,31,29);(5,8)(6,13)(7,18)(9,26)(10,24)(11,29)(12,14)(15,33)(16,32)(17,20)(19,35)(21,36)(22,25)(23,27)(28,31)(30,34)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,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, 363160);