#1043794

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3062 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)(3,4,7,8)(9,10,12,14)(11,15,16,13)(18,20)(19,23)(25,26,28,29)(27,31,32,30);(2,6)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,22,23,24)(26,29)(30,31);(2,6)(4,8)(11,16)(13,15)(18,20)(22,24)(25,27,28,32)(26,30,29,31);(1,3)(2,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,29)(31,32);(1,4)(2,3)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,24,23,22)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327625, #72327671, #72327703
maximal normal subgroup of	#72327625, #72327671, #72327703
maximal quotient of	#72328811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 950990);