#2607263

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

3164 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267028
maximal normal subgroup of	#72267028
maximal quotient of	#72267036

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2514459);