#7658126

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×#3514	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6640 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)(7,13)(11,14)(15,19)(18,20)(21,22,27,28)(23,26,30,25)(24,32,35,29)(31,33,36,34);(1,2,3,4)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(22,28)(23,30)(29,32)(31,36);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,23,27,30)(22,25,28,26)(24,31,35,36)(29,33,32,34);(21,24,27,35)(22,29,28,32)(23,31,30,36)(25,33,26,34);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,25,27,26)(22,23,28,30)(24,33,35,34)(29,31,32,36);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,26,27,25)(22,30,28,23)(24,34,35,33)(29,36,32,31)

Polycyclic presentation:

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

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, 7565322);