#420054

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1372 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:

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

Polycyclic presentation:

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

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, 327250);