#7715557

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×#3558	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7622753);