#95710

order	512
nilpotency class	3
isoclinism	isoclinic class of #95118
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fh,d^b=dh,e^b=eg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309101, #72309110, #72309164, …
maximal normal subgroup of	#72309101, #72309110, #72309164, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2906);