#466756

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23438	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1297 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ef,d^2=f,e^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^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373952);