#516686

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2530 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423882);