#7680633

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=hi,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, 7587829);