#556171

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1992 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=i,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,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, 463367);