#1186874

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1094070);