#423243

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1516 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 330439);