#420052

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1364 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:

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

Polycyclic presentation:

2^9|a^2=efi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,h^c=hi,f^d=fi,h^d=hi

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, 327248);