#2120746

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2076 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:

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

Polycyclic presentation:

2^9|a^2=fghi,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2027942);