#949238

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2070 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgh,d^2=i,e^2=g,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, 856434);