#918123

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

2845 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)(9,10,13,16)(11,12,15,14)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31);(5,6,7,8)(11,15)(12,14)(18,22)(19,26)(24,28)(27,32);(5,6)(7,8)(9,11,13,15)(10,14,16,12)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(1,2)(3,4)(9,12,13,14)(10,15,16,11)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fg,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 825319);