#7715823

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

6892 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)(8,16)(11,13)(14,20)(24,32)(28,33)(31,35)(34,36);(1,2)(3,4)(6,10)(11,13)(12,17)(18,19)(22,26)(23,29)(28,33)(31,35);(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7623019);