#7627612

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
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×#3511	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6628 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:

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

Polycyclic presentation:

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

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, 7534808);