#7712670

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×#3444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6854 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)(8,16)(12,17)(14,20)(18,19)(21,22)(23,28)(24,26)(25,27)(29,33)(30,35)(31,32)(34,36);(1,2)(3,4)(6,10)(11,13)(12,17)(18,19)(21,23,27,33)(22,28,25,29)(24,31,30,34)(26,32,35,36);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23)(22,29)(24,34)(25,28)(26,32)(27,33)(30,31)(35,36);(21,24)(22,26)(23,31)(25,35)(27,30)(28,32)(29,36)(33,34);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,25,27,22)(23,29,33,28)(24,35,30,26)(31,36,34,32);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,26,27,35)(22,30,25,24)(23,32,33,36)(28,34,29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,f^2=i,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, 7619866);