#154075

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3590 conjugacy classes of subgroups; only the 658 normal subgroups are shown.

normal subgroups only

This group has 3590 conjugacy classes of subgroups — too many to draw. Only its 658 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258449
maximal normal subgroup of	#72258449

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61271);