#7674865

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	#35358	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5390 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7582061);