#7695076

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	#52117	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4651 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)(9,10,13,11)(12,14,16,15)(17,18,22,23)(19,27,28,24)(20,25,30,29)(21,26,31,32);(1,2)(3,6)(4,5)(7,8)(17,18)(19,24)(20,25)(21,26)(22,23)(27,28)(29,30)(31,32);(10,11)(14,15)(18,23)(19,28)(25,29)(26,32);(9,11,13,10)(12,15,16,14)(17,19,22,28)(18,24,23,27)(20,26,30,32)(21,29,31,25);(9,12,13,16)(10,14,11,15)(17,20)(18,25)(19,26)(21,24)(22,30)(23,29)(27,31)(28,32);(1,3,4,7)(2,6,5,8)(17,21)(18,26)(19,29)(20,27)(22,31)(23,32)(24,30)(25,28)

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7602272);