#393294

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=efi,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,c^b=ci,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300490);