#7712647

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×#3561	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6570 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)(9,17)(13,18)(14,19)(16,20)(21,22,27,32)(23,30,25,28)(24,29,33,34)(26,36,35,31);(1,2,3,4)(5,6,10,11)(7,8,12,15)(9,13,17,18)(14,16,19,20)(21,23,27,25)(22,28,32,30)(24,26,33,35)(29,31,34,36);(6,11)(8,15)(13,18)(16,20)(21,22)(23,30)(24,34)(25,28)(26,31)(27,32)(29,33)(35,36);(5,7,10,12)(6,8,11,15)(9,14,17,19)(13,16,18,20)(21,24)(22,29)(23,26)(25,35)(27,33)(28,31)(30,36)(32,34);(5,8)(6,12)(7,11)(9,16)(10,15)(13,19)(14,18)(17,20)(21,25,27,23)(22,30,32,28)(24,35,33,26)(29,36,34,31);(5,9,10,17)(6,13,11,18)(7,14,12,19)(8,16,15,20)(21,26,27,35)(22,31,32,36)(23,33,25,24)(28,34,30,29)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=hi,e^2=h,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, 7619843);