#7632195

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4963 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)(10,11)(15,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31);(1,2,5,4)(3,6,7,8)(9,10,14,11)(12,15,13,16)(17,19)(18,22)(20,30)(21,29)(23,26)(24,32)(25,31)(27,28);(1,3,5,7)(2,6,4,8)(9,11,14,10)(12,16,13,15)(17,20,23,21)(18,24,27,25)(19,29,26,30)(22,31,28,32);(9,12,14,13)(10,15,11,16)(17,21,23,20)(18,25,27,24)(19,29,26,30)(22,31,28,32);(9,12,14,13)(10,15,11,16)(17,22,23,28)(18,26,27,19)(20,31,21,32)(24,30,25,29);(1,4,5,2)(3,8,7,6)(9,13,14,12)(10,16,11,15)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539391);