#7710611

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	#36706	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6862 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)(9,10)(11,14)(12,17)(15,18)(19,20)(25,33)(29,34)(30,35)(32,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,22,26,27)(23,24,28,31)(25,29,33,34)(30,32,35,36);(6,10)(7,14)(12,16)(15,20)(22,27)(24,31)(29,34)(32,36);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,23,26,28)(22,24,27,31)(25,30,33,35)(29,32,34,36);(5,7,9,14)(6,11,10,13)(8,15,17,20)(12,18,16,19)(21,24)(22,28)(23,27)(25,32)(26,31)(29,35)(30,34)(33,36);(5,8)(6,12)(7,15)(9,17)(10,16)(11,18)(13,19)(14,20)(21,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,35)(31,36)

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,e^2=h,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, 7617807);