#945704

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2540	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3834 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 852900);