#1452058

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	#14762	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2539 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gi,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1359254);