#1394088

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2617	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3666 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:

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

Polycyclic presentation:

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

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, 1301284);