#7918994

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

6575 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)(6,10)(7,13)(12,18)(14,15)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,17)(14,19)(15,20)(16,18);(5,6,9,10)(7,11,13,17)(8,12,16,18)(14,19,15,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,6)(7,11)(8,12)(9,10)(13,17)(14,19)(15,20)(16,18)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,7)(6,11)(8,15)(9,13)(10,17)(12,20)(14,16)(18,19)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35);(5,8,9,16)(6,12,10,18)(7,14,13,15)(11,19,17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7826190);