#1393501

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2635	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(11,13)(12,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31);(6,8)(9,10,14,16)(11,15,13,12)(18,23)(19,25)(24,31)(26,32);(5,6)(7,8)(9,11,14,13)(10,15,16,12)(17,19,22,25)(18,20,23,28)(21,26,30,32)(24,27,31,29);(1,2)(3,4)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,29)(22,28)(23,25)(24,32)(26,31)(27,30);(9,13,14,11)(10,12,16,15)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)

Polycyclic presentation:

2^9|b^2=g,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, 1300697);