#441220

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1213 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)(7,17,21,12)(8,22,26,31)(9,16)(10,14)(11,15)(13,24,32,25)(18,34)(19,28,20,29)(23,33)(27,35)(30,36);(5,7,10,19,11,21,9,20)(6,12,14,29,15,17,16,28)(8,23,24,36,26,35,25,18)(13,30,22,27,32,34,31,33);(2,4)(5,8,11,26)(6,13)(7,18,21,36)(9,24,10,25)(12,27)(14,31)(15,32)(16,22)(17,33)(19,35,20,23)(28,34)(29,30);(6,14,15,16)(12,28,17,29)(13,31,32,22)(27,34,33,30)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=ei,d^2=e,f^2=i,g^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,f^b=fi,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, 348416);