#238085

order	512
nilpotency class	3
isoclinism	isoclinic class of #238082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1462 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235846, #72261117, #72261134, …
maximal normal subgroup of	#72235846, #72261117, #72261134, …
maximal quotient of	#72303548

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145281);