#648102

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 555298);