#7658697

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,6,11,10)(7,12,17,16)(8,18,9,13)(14,20,15,19)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,34,36,35);(1,2)(3,4)(5,7,11,17)(6,12,10,16)(8,14,9,15)(13,19,18,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,8,11,9)(6,13,10,18)(7,14,17,15)(12,19,16,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,7)(6,12)(8,14)(9,15)(10,16)(11,17)(13,19)(18,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36);(5,9,11,8)(6,13,10,18)(7,15,17,14)(12,19,16,20);(5,10,11,6)(7,16,17,12)(8,18,9,13)(14,20,15,19)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565893);