#515666

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
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

2382 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)(5,6)(7,12)(8,9)(10,11)(13,14,17,18)(15,16,19,20)(22,25)(26,28);(6,11)(7,8)(14,18)(15,19)(22,25)(26,28);(1,3,4,2)(5,7,10,8)(6,12,11,9)(13,14)(15,16)(17,18)(19,20)(21,22)(23,26)(24,25)(27,28);(5,8,10,7)(6,9,11,12)(13,15)(14,16)(17,19)(18,20);(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=gh,e^2=fh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=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, 422862);