#157549

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

3168 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fh,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278387, #72278397, #72278412, …
maximal normal subgroup of	#72278387, #72278397, #72278412, …
maximal quotient of	#72299901

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 64745);