#7627609

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6298 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534805);