#1145465

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2414 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1052661);