#7627327

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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×#535	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6700 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)(5,6,11,10)(7,14,9,12)(8,16,18,13)(15,19,17,20)(21,22)(23,27)(24,30)(25,29)(26,31)(28,35)(32,34)(33,36);(5,7,11,9)(6,12,10,14)(8,15,18,17)(13,19,16,20)(21,23,26,29)(22,27,31,25)(24,33,35,32)(28,34,30,36);(1,3,4,2)(5,8)(6,13)(7,15)(9,17)(10,16)(11,18)(12,19)(14,20)(21,24)(22,28)(23,32)(25,34)(26,35)(27,36)(29,33)(30,31);(5,8)(6,13)(7,15)(9,17)(10,16)(11,18)(12,19)(14,20)(21,25,26,27)(22,29,31,23)(24,34,35,36)(28,33,30,32);(5,9,11,7)(6,12,10,14)(8,17,18,15)(13,19,16,20)(21,24)(22,30)(23,33)(25,34)(26,35)(27,36)(28,31)(29,32);(5,10,11,6)(7,14,9,12)(8,16,18,13)(15,20,17,19)(21,22)(23,27)(24,28)(25,29)(26,31)(30,35)(32,36)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534523);