#518029

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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

2790 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=fh,e^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=ef

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, 425225);