#423865

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1319 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 331061);