#1462727

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fhi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

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, 1369923);