#1248977

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1156173);