#652580

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2232 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)(7,9)(8,11)(10,12)(13,14,17,16)(15,18,20,19)(21,22,24,26)(23,27,28,25);(5,7,10,8)(6,11,12,9)(14,16)(18,19);(5,6)(7,9)(8,11)(10,12)(13,15,17,20)(14,18,16,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,8,10,7)(6,11,12,9)(21,23,24,28)(22,25,26,27);(5,9,10,11)(6,7,12,8)(13,16,17,14)(15,19,20,18)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=fh,d^2=gh,e^2=f,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, 559776);