#1265216

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15412	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2705 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=f,e^2=gh,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, 1172412);