#438349

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=fi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 345545);