#7678288

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	#35223	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5162 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)(3,5,8,7)(10,11)(14,15)(17,18)(19,26)(20,25)(21,22)(23,24)(27,32)(28,31)(29,30);(9,10,13,11)(12,14,16,15)(17,19)(18,22)(20,27)(21,23)(24,26)(25,30)(28,29)(31,32);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,14,16,15)(18,24)(19,21)(20,28)(30,32);(1,3)(2,5)(4,8)(6,7)(9,12,13,16)(10,14,11,15)(17,20,23,28)(18,25,24,31)(19,27,21,29)(22,30,26,32);(17,21,23,19)(18,22,24,26)(20,29,28,27)(25,30,31,32);(17,22)(18,19)(20,30)(21,24)(23,26)(25,27)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^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, 7585484);