#7627587

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×#3553	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6390 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)(7,12)(8,11)(9,13)(10,15)(14,19)(16,17)(18,20)(21,22)(23,32)(24,26)(25,28)(27,31)(29,34)(30,35)(33,36);(5,6)(7,12)(8,16)(9,13)(10,15)(11,17)(14,18)(19,20)(21,23,28,34)(22,29,25,32)(24,33,35,31)(26,27,30,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,24,28,35)(22,30,25,26)(23,33,34,31)(27,29,36,32);(5,8,9,17)(6,11,13,16)(7,14,15,20)(10,18,12,19)(21,25,28,22)(23,29,34,32)(24,26,35,30)(27,33,36,31);(1,3,4,2)(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,18)(17,20)(21,26)(22,24)(23,27)(25,35)(28,30)(29,33)(31,32)(34,36);(5,6)(7,10)(8,11)(9,13)(12,15)(14,18)(16,17)(19,20)(21,27)(22,31)(23,30)(24,29)(25,33)(26,34)(28,36)(32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=i,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, 7534783);