#7678008

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×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5022 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gh,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, 7585204);