#7626681

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×#3505	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6964 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)(5,6,10,13)(7,14,8,11)(9,17,18,12)(15,19,16,20)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,32)(35,36);(1,3,4,2)(5,7,10,8)(6,11,13,14)(9,15,18,16)(12,19,17,20);(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,19)(17,18)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36);(5,6)(7,11)(8,14)(9,12)(10,13)(15,19)(16,20)(17,18)(21,22)(23,24)(25,28)(26,29)(27,30)(31,32)(33,34)(35,36);(5,8,10,7)(6,11,13,14)(9,16,18,15)(12,19,17,20)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,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, 7533877);