#2760695

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 2097152
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2416 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)(10,12)(11,15)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32);(6,8)(9,10,13,12)(11,16,15,14)(19,23)(20,26)(27,31)(28,32);(5,6)(7,8)(11,15)(14,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32);(1,2)(3,4)(9,11,13,15)(10,14,12,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32);(9,12,13,10)(11,14,15,16)(17,21,22,30)(18,24,25,29)(19,27,23,31)(20,28,26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2667891);