#2126274

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2317 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)(5,6,8,10)(7,11,12,9)(14,18)(16,19)(21,22,25,26)(23,24,28,27);(7,12)(9,11)(13,14,17,18)(15,16,20,19)(22,26)(23,28);(5,7)(6,9)(8,12)(10,11)(13,15,17,20)(14,19,18,16)(21,23,25,28)(22,27,26,24);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,15)(17,19)(18,20);(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24,25,27)(22,23,26,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=f,e^2=fgh,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, 2033470);