#400989

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1860 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=i,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,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, 308185);