#1293524

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2555	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1200720);