#513611

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂²	C₂³×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420807);