#517608

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^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, 424804);