#7627615

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=h,e^2=i,f^2=g,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, 7534811);