#927498

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=gi,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, 834694);