#7681798

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#34935	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5096 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)(6,7)(10,11)(14,15)(17,18,23,20)(19,24,26,21)(22,28,30,25)(27,29,32,31);(9,10,13,11)(12,14,16,15)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32);(1,2)(3,7)(4,5)(6,8)(9,11,13,10)(12,14,16,15)(19,26)(21,24)(27,32)(29,31);(1,3,4,8)(2,6,5,7)(9,12,13,16)(10,14,11,15)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31);(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31);(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,f^2=i,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, 7588994);