#313100

order	512
nilpotency class	3
isoclinism	isoclinic class of #264363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=efhi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,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, 220296);