#517616

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×2₋¹⁺⁴	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

17742 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,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, 424812);