#7627529

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×#3559	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,e^2=i,f^2=h,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, 7534725);