#442709

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

838 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)(5,6,9,13,10,14,24,15)(7,16,19,26,20,30,18,11)(8,21,35,28,23,12,22,29)(17,31,32,25,34,36,33,27);(5,7,10,20)(6,11,14,26)(8,17)(9,18,24,19)(12,27)(13,30,15,16)(21,25)(22,32)(23,34)(28,31)(29,36)(33,35);(5,8,9,22,10,23,24,35)(6,12,13,28,14,21,15,29)(7,17,19,33,20,34,18,32)(11,25,16,31,26,27,30,36);(2,4)(6,13,14,15)(7,18,20,19)(11,26)(12,28,21,29)(16,30)(17,32,34,33)(25,27)(31,36)

Polycyclic presentation:

2^9|a^2=efg,b^2=fgi,c^2=e,d^2=f,e^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,g^c=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, 349905);