#1393453

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

3748 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)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27);(10,12)(14,19)(15,21)(20,27)(22,28);(1,2,4,3)(9,10)(11,12)(13,15,18,21)(14,16,19,24)(17,22,26,28)(20,23,27,25);(5,6)(7,8)(13,16)(14,15)(17,25)(18,24)(19,21)(20,28)(22,27)(23,26);(1,3,4,2)(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,28)(24,25)

Polycyclic presentation:

2^9|c^2=fg,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, 1300649);