#7631965

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	#36214	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5408 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,f^2=g,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, 7539161);