#7711926

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	#36624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4888 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239143, #72239150
maximal normal subgroup of	#72239143, #72239150

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619122);