#396114

order	512
nilpotency class	3
isoclinism	isoclinic class of #396113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1390 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)(5,6,9,15,11,12,10,25)(7,16,19,26,20,27,17,13)(8,21,22,36,24,14,23,29)(18,32,33,28,34,30,31,35);(6,12)(7,17)(9,10)(13,27)(14,21)(16,26)(18,31)(19,20)(22,23)(28,32)(30,35)(33,34);(2,4)(5,7,11,20)(6,13,12,26)(8,18,24,34)(9,17,10,19)(14,28,21,35)(15,27,25,16)(22,31,23,33)(29,32,36,30);(5,8)(6,14,12,21)(7,18)(9,22)(10,23)(11,24)(13,28,26,35)(15,29,25,36)(16,30,27,32)(17,31)(19,33)(20,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303310);