#7711537

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239250, #72239255
maximal normal subgroup of	#72239250, #72239255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618733);