#2244525

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

3125 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,16,12,14)(17,18,22,24)(19,20,21,23)(25,26,28,29)(27,30,31,32);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,22,21)(18,23,24,20)(26,29)(30,32);(3,7)(6,8)(11,12)(14,16)(18,24)(19,21)(25,27,28,31)(26,30,29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,14,15,16)(17,21,22,19)(18,20,24,23)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327829, #72327908, #72327910, …
maximal normal subgroup of	#72327829, #72327908, #72327910, …
maximal quotient of	#72328850, #72328853

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2151721);