#1195210

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2671	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3141 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=fg,d^2=fhi,e^2=fg,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, 1102406);