#353481

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2090 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)(13,14,17,21,19,23,18,22)(15,16,20,27,24,26,25,28);(9,10,12,11)(13,15,18,25,19,24,17,20)(14,16,22,28,23,26,21,27);(2,4)(9,11,12,10)(13,16)(14,20)(15,22)(17,27)(18,28)(19,26)(21,24)(23,25);(15,24)(16,26)(20,25)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^b=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260677);