#1294198

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2839 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(10,13)(11,15)(17,18,22,23)(19,26,25,21)(20,28,30,24)(27,31,32,29);(5,6,7,8)(10,13)(14,16)(18,23)(19,25)(24,28)(27,32);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,25)(18,21,23,26)(20,27,30,32)(24,31,28,29);(1,2,3,4)(17,20,22,30)(18,24,23,28)(19,27,25,32)(21,31,26,29);(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,23)(20,29)(22,26)(24,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280328, #72280375
maximal normal subgroup of	#72280328, #72280375

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1201394);