#7826529

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318151, #72318169, #72318513, …
maximal normal subgroup of	#72318151, #72318169, #72318513, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7733725);