#770065

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3066 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)(6,7)(11,15)(13,16)(17,18,22,23)(19,24,27,32)(20,28,29,25)(21,30,31,26);(9,10,12,14)(11,13,15,16)(18,23)(20,29)(24,32)(26,30);(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,26)(21,28)(22,27)(23,32)(25,31)(29,30);(1,2,3,4)(5,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,20,22,29)(18,25,23,28)(19,26,27,30)(21,24,31,32);(17,21,22,31)(18,26,23,30)(19,25,27,28)(20,24,29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 677261);