#424328

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1497 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 331524);