#449590

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1974 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)(7,12)(8,19,22,13)(9,14)(10,25)(11,15)(16,27)(17,31,21,28)(18,33)(20,30,35,36)(23,32,26,34)(24,29);(5,7,11,9)(6,12,15,14)(8,20,22,35)(10,24,27,18)(13,30,19,36)(16,33,25,29)(17,26,21,23)(28,34,31,32);(5,8)(6,13)(7,17)(9,21)(10,26)(11,22)(12,28)(14,31)(15,19)(16,34)(18,35)(20,24)(23,27)(25,32)(29,36)(30,33);(2,4)(5,9,11,7)(6,14,15,12)(8,21,22,17)(10,18,27,24)(13,31,19,28)(16,29,25,33)(20,26,35,23)(30,34,36,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356786);