#7633413

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

7116 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:

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

Polycyclic presentation:

2^9|e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi

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, 7540609);