#422760

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60645	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1911 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:

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

Polycyclic presentation:

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

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, 329956);