#251856

order	512
nilpotency class	3
isoclinism	isoclinic class of #251827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#46387	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1188 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1188 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240633, #72240640, #72251343, …
maximal normal subgroup of	#72240633, #72240640, #72251343, …
maximal quotient of	#72251588

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159052);