#7627252

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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

6876 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)(8,14)(9,13)(10,15)(12,19)(16,18)(17,20)(22,24)(23,28)(29,32)(31,36);(5,7,10,13)(6,11,15,9)(8,17,19,16)(12,18,14,20)(21,22,27,24)(23,25,28,30)(26,29,35,32)(31,33,36,34);(5,8)(6,12)(7,16)(9,18)(10,19)(11,20)(13,17)(14,15)(21,23)(22,25)(24,30)(26,31)(27,28)(29,33)(32,34)(35,36);(5,9,10,11)(6,13,15,7)(8,18,19,20)(12,17,14,16)(21,24,27,22)(23,30,28,25)(26,32,35,29)(31,34,36,33);(5,8)(6,14)(7,17)(9,18)(10,19)(11,20)(12,15)(13,16)(21,25,27,30)(22,28,24,23)(26,34,35,33)(29,31,32,36);(5,6)(7,11)(8,12)(9,13)(10,15)(14,19)(16,20)(17,18)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,36)(30,34)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534448);