#7627653

order	512
nilpotency class	2
isoclinism	isoclinic class of #62972
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×#3543	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7242 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)(7,10)(8,11)(9,14)(12,17)(13,15)(16,19)(18,20)(21,22,26,24)(23,29,30,27)(25,28,33,32)(31,35,36,34);(6,7)(8,15)(9,17)(18,19)(23,30)(25,33)(27,29)(28,32);(6,7)(8,15)(12,14)(16,20)(22,24)(27,29)(28,32)(34,35);(5,7,10,6)(8,13,15,11)(9,14,17,12)(16,19,20,18)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36);(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,24,26,22)(23,29,30,27)(25,32,33,28)(31,35,36,34);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300320
maximal normal subgroup of	#72300320
maximal quotient of	#72300412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534849);