#7626639

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	C₂×#3475	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6746 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)(7,11)(8,16)(9,13)(10,15)(12,17)(14,18)(19,20)(22,27)(23,29)(25,33)(32,35);(21,22)(23,28)(24,29)(25,30)(26,27)(31,35)(32,36)(33,34);(5,7)(6,10)(8,14)(9,15)(11,13)(12,18)(16,19)(17,20)(21,23)(22,24)(25,31)(26,29)(27,28)(30,32)(33,36)(34,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,19)(16,18)(17,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,34)(33,36);(5,6)(7,10)(8,12)(9,13)(11,15)(14,18)(16,17)(19,20)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(5,8,9,17)(6,12,13,16)(7,14,15,20)(10,18,11,19)(21,25,26,33)(22,30,27,34)(23,31,29,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=i,e^2=g,f^2=h,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, 7533835);