#517983

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

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

4164 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^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, 425179);