#7626821

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
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	C₂²×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6858 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)(5,6,11,9)(7,14,8,12)(10,17,18,13)(15,19,16,20)(21,22,27,31)(23,30,34,28)(24,26,25,29)(32,33,35,36);(21,23)(22,28)(24,35)(25,32)(26,33)(27,34)(29,36)(30,31);(1,3,4,2)(5,7,11,8)(6,12,9,14)(10,15,18,16)(13,19,17,20)(21,24,27,25)(22,29,31,26)(23,32,34,35)(28,33,30,36);(5,8,11,7)(6,12,9,14)(10,16,18,15)(13,19,17,20)(21,25,27,24)(22,26,31,29)(23,32,34,35)(28,33,30,36);(5,9,11,6)(7,14,8,12)(10,17,18,13)(15,20,16,19)(21,23)(22,30)(24,32)(25,35)(26,33)(27,34)(28,31)(29,36);(5,10)(6,13)(7,15)(8,16)(9,17)(11,18)(12,19)(14,20)(21,26)(22,24)(23,33)(25,31)(27,29)(28,35)(30,32)(34,36)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534017);