#156496

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=efg,c^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222731, #72222738, #72222803
maximal normal subgroup of	#72222731, #72222738, #72222803

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63692);