#898732

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2768 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)(7,11)(10,12)(13,14,17,16)(15,19,20,18)(21,22,25,26)(23,24,28,27);(1,2,4,3)(5,6,9,8)(7,10,11,12)(15,20)(18,19)(22,26)(24,27);(13,15)(14,18)(16,19)(17,20)(21,23)(22,27)(24,26)(25,28);(1,2,4,3)(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,24,25,27)(22,23,26,28);(1,3,4,2)(5,8,9,6)(7,12,11,10)(13,16,17,14)(15,19,20,18)(21,23)(22,27)(24,26)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 805928);