#6788251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#31057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 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)(17,18)(19,26)(20,23)(21,25)(22,30)(24,28)(27,29)(31,32);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,24)(21,25)(22,31)(23,28)(26,29)(30,32);(2,5)(6,8)(10,13)(14,16)(17,19,28,29,21,31,20,30)(18,22,23,32,25,27,24,26);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,18)(19,22)(20,24)(21,25)(23,28)(26,30)(27,31)(29,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,28)(18,23,25,24)(19,29,31,30)(22,26,27,32)

Polycyclic presentation:

2^9|c^2=fgi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313330, #72313405, #72314027
maximal normal subgroup of	#72313330, #72313405, #72314027

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6695447);