#7626588

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×#555	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7020 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=g,e^2=g,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, 7533784);