#8092464

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304025
maximal normal subgroup of	#72304025
maximal quotient of	#72304096, #72304123

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7999660);