#7644740

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×#3505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7004 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)(8,16)(12,19)(14,20)(17,18)(22,27)(23,28)(25,33)(31,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,15)(14,18)(16,19)(17,20)(21,22,26,27)(23,24,28,32)(25,29,33,34)(30,31,35,36);(5,7)(6,10)(8,14)(9,15)(11,13)(12,17)(16,20)(18,19)(21,22,26,27)(23,24,28,32)(25,29,33,34)(30,31,35,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,35)(34,36);(5,6)(7,10)(8,12)(9,13)(11,15)(14,17)(16,19)(18,20)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,34)(33,36);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,29)(23,30)(24,31)(26,33)(27,34)(28,35)(32,36)

Polycyclic presentation:

2^9|b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7551936);