#424248

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1501 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=e,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,g^b=gi,g^c=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, 331444);