#7627316

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×#3521	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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,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, 7534512);