#423624

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1562 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)(7,16)(8,13)(9,22)(10,26)(11,15)(12,19)(14,28)(17,33)(18,34)(20,32)(21,24)(23,31)(25,36)(27,30)(29,35);(5,7)(6,12)(8,21)(9,24)(10,27)(11,20)(13,18)(14,32)(15,30)(16,26)(17,23)(19,28)(22,29)(25,33)(31,35)(34,36);(5,8,11,23)(6,13,15,31)(7,17)(9,14,25,26)(10,22,28,36)(12,18)(16,24)(19,29)(20,21)(27,34)(30,35)(32,33);(2,4)(5,9,10,13,11,25,28,31)(6,8,14,22,15,23,26,36)(7,18,19,24,20,35,27,33)(12,29,16,21,30,34,32,17)

Polycyclic presentation:

2^9|c^2=fi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=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, 330820);