#7626685

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×#3515	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6472 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)(7,14)(8,11)(9,17)(10,13)(12,18)(15,16)(19,20)(21,22)(23,24)(25,29)(26,27)(28,32)(30,31)(33,34)(35,36);(1,3,4,2)(22,27)(23,28)(29,34)(30,35);(5,7,10,11)(6,8,13,14)(9,15,18,19)(12,16,17,20)(21,23,26,28)(22,24,27,32)(25,30,33,35)(29,31,34,36);(5,6)(7,8)(9,12)(10,13)(11,14)(15,16)(17,18)(19,20)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,34)(33,36);(5,8)(6,11)(7,13)(9,16)(10,14)(12,19)(15,17)(18,20);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)(21,25,26,33)(22,29,27,34)(23,30,28,35)(24,31,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,f^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, 7533881);