#213880

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#37232	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1280 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240626, #72240627, #72240628, …
maximal normal subgroup of	#72240626, #72240627, #72240628, …
maximal quotient of	#72251455, #72251497

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121076);