#7626426

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
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×#3514	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6830 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)(7,9)(8,12)(10,11)(13,17)(14,16)(15,18)(19,20)(21,22)(23,32)(24,25)(26,36)(27,29)(28,33)(30,35)(31,34);(8,15)(12,18)(14,19)(16,20)(21,23,27,33)(22,28,29,32)(24,34,35,26)(25,36,30,31);(1,3,4,2)(6,11)(7,13)(12,18)(14,19)(22,29)(23,33)(24,35)(31,36);(5,7,10,13)(6,9,11,17)(8,14,15,19)(12,16,18,20)(21,24)(22,30)(23,34)(25,29)(26,33)(27,35)(28,31)(32,36);(5,8,10,15)(6,12,11,18)(7,14,13,19)(9,16,17,20)(21,25,27,30)(22,24,29,35)(23,31,33,36)(26,32,34,28);(5,9)(6,13)(7,11)(8,16)(10,17)(12,19)(14,18)(15,20)(21,26,27,34)(22,31,29,36)(23,24,33,35)(25,32,30,28)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533622);