#7732040

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×#3460	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6754 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)(5,6,11,13)(7,12,9,19)(8,14,17,10)(15,16,18,20);(1,2)(3,4)(5,7,11,9)(6,12,13,19)(8,15,17,18)(10,20,14,16)(22,26)(24,32)(27,29)(30,36);(6,13)(7,9)(8,17)(16,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8)(6,14)(7,15)(9,18)(10,13)(11,17)(12,16)(19,20);(5,9,11,7)(6,12,13,19)(8,18,17,15)(10,20,14,16)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,10,11,14)(6,8,13,17)(7,16,9,20)(12,18,19,15)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7639236);