#7627475

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×#3533	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6742 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)(7,12)(8,11)(9,13)(10,15)(14,19)(16,17)(18,20)(21,22,28,26)(23,31,27,29)(24,34,30,25)(32,36,35,33);(5,6)(7,12)(8,16)(9,13)(10,15)(11,17)(14,18)(19,20)(21,23,28,27)(22,29,26,31)(24,32,30,35)(25,36,34,33);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,24)(22,25)(23,32)(26,34)(27,35)(28,30)(29,36)(31,33);(5,8)(6,11)(7,14)(9,17)(10,18)(12,19)(13,16)(15,20)(21,25,28,34)(22,30,26,24)(23,33,27,36)(29,32,31,35);(1,3,4,2)(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,18)(17,20)(21,26,28,22)(23,31,27,29)(24,34,30,25)(32,33,35,36);(5,6)(7,10)(8,11)(9,13)(12,15)(14,18)(16,17)(19,20)(21,27,28,23)(22,29,26,31)(24,35,30,32)(25,36,34,33)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=i,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, 7534671);