#917219

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3839 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)(5,6,7,8)(13,14)(15,20)(16,22)(17,25)(18,19)(21,26)(23,24)(27,28);(5,6,7,8)(10,12)(14,19)(15,22)(21,25)(23,28);(5,6)(7,8)(9,10)(11,12)(13,15,18,22)(14,20,19,16)(17,23,26,28)(21,27,25,24);(1,2)(3,4)(5,6)(7,8)(13,16,18,20)(14,15,19,22)(17,24,26,27)(21,23,25,28);(13,17)(14,21)(15,23)(16,24)(18,26)(19,25)(20,27)(22,28)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^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, 824415);