#6388161

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295357);