#945527

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

2602 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgh,c^2=fghi,d^2=g,e^2=fgh,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, 852723);