#7920129

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

7039 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)(8,11)(9,14)(10,12)(15,20)(16,18)(17,19)(21,22,26,28)(23,27,29,24)(25,30,34,35)(31,33,36,32);(1,2)(3,4)(5,7,10,8)(6,11,12,13)(9,15,17,16)(14,18,19,20)(21,23,26,29)(22,27,28,24)(25,31,34,36)(30,33,35,32);(6,12)(11,13)(14,19)(18,20)(22,28)(23,29)(30,35)(31,36);(5,6)(7,11)(8,13)(9,14)(10,12)(15,18)(16,20)(17,19)(21,22)(23,27)(24,29)(25,30)(26,28)(31,33)(32,36)(34,35);(5,8,10,7)(6,13,12,11)(9,16,17,15)(14,20,19,18)(21,24,26,27)(22,29,28,23)(25,33,34,32)(30,31,35,36);(5,9,10,17)(6,14,12,19)(7,15,8,16)(11,18,13,20)(21,25,26,34)(22,30,28,35)(23,31,29,36)(24,32,27,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=h,f^2=h,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, 7827325);