#2844108

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2123 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)(3,8)(4,5)(6,7)(10,13)(11,12)(17,18,21,20)(19,23,24,22)(25,26,28,30)(27,31,32,29);(2,7)(5,8)(9,10)(11,15)(12,16)(13,14)(18,20)(22,23)(25,26,28,30)(27,29,32,31);(1,3,6,4)(2,8,7,5)(9,11,14,12)(10,15,13,16)(17,18,21,20)(19,22,24,23)(25,27)(26,29)(28,32)(30,31);(1,4,6,3)(2,8,7,5)(9,12,14,11)(10,15,13,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,5,6,8)(2,4,7,3)(9,13,14,10)(11,16,12,15)(17,20,21,18)(19,22,24,23)(25,27)(26,29)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=f,d^2=fh,e^2=f,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, 2751304);