#980477

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2826 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=f,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 887673);