#7679777

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4438 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)(10,13)(14,16)(18,23)(21,32)(24,31)(25,29);(9,10,12,13)(11,14,15,16)(17,18,22,23)(19,24,28,31)(20,29,30,25)(21,26,32,27);(1,3,5,8)(2,6,4,7)(18,23)(20,30)(24,31)(26,27);(1,4,5,2)(3,6,8,7)(9,11,12,15)(10,14,13,16)(17,19,22,28)(18,24,23,31)(20,27,30,26)(21,25,32,29);(17,20)(18,25)(19,27)(21,31)(22,30)(23,29)(24,32)(26,28);(17,21,22,32)(18,26,23,27)(19,25,28,29)(20,31,30,24)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,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, 7586973);