#2306511

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 2097152
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2486 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)(11,15)(12,16)(13,14)(17,18,22,23)(19,24,27,31)(20,21,26,25)(28,30,32,29);(5,6,7,8)(9,11)(10,12)(13,16)(14,15)(18,23)(19,27)(20,26)(29,30);(5,6)(7,8)(10,14)(12,15)(17,19)(18,24)(20,28)(21,30)(22,27)(23,31)(25,29)(26,32);(1,2,3,4)(9,11)(10,12)(13,16)(14,15)(17,20,22,26)(18,25,23,21)(19,28,27,32)(24,29,31,30);(9,12)(10,11)(13,15)(14,16)(17,21,22,25)(18,26,23,20)(19,29,27,30)(24,28,31,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2213707);