#517958

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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

3118 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)(7,10)(8,11)(9,12)(13,14)(15,16)(17,20)(18,19)(21,22,25,28)(23,24,27,26);(5,7)(6,9)(8,12)(10,11)(13,15,17,19)(14,18,20,16)(21,23)(22,26)(24,28)(25,27);(1,2,3,4)(13,16)(14,19)(15,20)(17,18)(21,22,25,28)(23,24,27,26);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28);(5,6)(7,9)(8,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,24)(22,27)(23,28)(25,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,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, 425154);