#7711554

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×#3535	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6808 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)(7,13)(8,16)(9,12)(10,15)(11,17)(14,18)(19,20)(25,33)(28,32)(30,36)(34,35);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,17)(19,20)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(5,6,9,12)(7,13,15,10)(8,11,17,16)(14,19,20,18)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,17)(19,20);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,24,26,22)(23,27,31,29)(25,32,33,28)(30,34,36,35);(5,8)(6,11)(7,14)(9,17)(10,18)(12,16)(13,19)(15,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618750);