#2844149

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2751345);