#1126598

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2772 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=gi,c^2=fgh,d^2=f,e^2=fgh,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.

Relationships

maximal subgroup of	#72326812, #72326825, #72326911
maximal normal subgroup of	#72326812, #72326825, #72326911

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1033794);