#415428

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322624);