#7627323

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×#3524	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6696 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)(5,6,12,10)(7,8,13,9)(11,14,19,18)(15,16,20,17)(21,22)(23,27)(24,30)(25,29)(26,31)(28,35)(32,34)(33,36);(5,7)(6,9)(8,10)(11,15)(12,13)(14,17)(16,18)(19,20)(21,23,26,29)(22,27,31,25)(24,33,35,32)(28,34,30,36);(1,3,4,2)(5,8,12,9)(6,7,10,13)(11,16,19,17)(14,15,18,20)(21,24)(22,28)(23,32)(25,34)(26,35)(27,36)(29,33)(30,31);(5,9,12,8)(6,13,10,7)(11,17,19,16)(14,20,18,15)(21,25,26,27)(22,29,31,23)(24,34,35,36)(28,33,30,32);(5,10,12,6)(7,8,13,9)(11,14,19,18)(15,17,20,16)(21,24)(22,30)(23,33)(25,34)(26,35)(27,36)(28,31)(29,32);(5,11,12,19)(6,14,10,18)(7,15,13,20)(8,16,9,17)(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=gh,c^2=i,d^2=h,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, 7534519);