#434833

order	512
nilpotency class	3
isoclinism	isoclinic class of #419843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

963 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)(3,4)(5,6,9,23,11,14,10,15)(7,12,16,28,18,27,24,25)(8,19,20,13,22,30,21,29)(17,32,33,36,34,35,31,26);(7,16,18,24)(12,25,27,28)(17,33,34,31)(26,35,36,32);(5,7)(6,12,14,27)(8,17)(9,24)(10,16)(11,18)(13,26,29,36)(15,28,23,25)(19,32,30,35)(20,31)(21,33)(22,34);(2,4)(5,8,11,22)(6,13,14,29)(7,17,18,34)(9,21,10,20)(12,26,27,36)(15,30,23,19)(16,31,24,33)(25,32,28,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=e,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=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, 342029);