#945624

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

3696 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)(6,7)(13,14,19,22)(15,20,16,17)(18,26,27,21)(23,25,24,28);(6,7)(10,12)(13,15,19,16)(14,17,22,20)(18,23,27,24)(21,25,26,28);(5,6,8,7)(9,10)(11,12)(13,16,19,15)(14,20,22,17)(18,24,27,23)(21,28,26,25);(1,2,3,4)(5,7,8,6)(13,17,19,20)(14,16,22,15)(18,25,27,28)(21,24,26,23);(13,18,19,27)(14,21,22,26)(15,23,16,24)(17,25,20,28)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=i,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, 852820);