#7848997

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	#52165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6491 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)(23,25)(28,32)(33,35)(34,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,15)(16,20)(17,18)(21,22,27,31)(23,32,25,28)(24,29,30,26)(33,34,35,36);(6,9)(11,14)(12,17)(19,20)(21,23,27,25)(22,28,31,32)(24,35,30,33)(26,36,29,34);(5,7,10,15)(6,11,9,14)(8,16,18,13)(12,20,17,19)(21,24,27,30)(22,29,31,26)(23,33,25,35)(28,36,32,34);(5,8,10,18)(6,12,9,17)(7,13,15,16)(11,19,14,20)(21,25,27,23)(22,28,31,32)(24,33,30,35)(26,36,29,34);(5,9,10,6)(7,14,15,11)(8,17,18,12)(13,20,16,19)(21,26,27,29)(22,30,31,24)(23,34,25,36)(28,35,32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7756193);