#95213

order	512
nilpotency class	3
isoclinism	isoclinic class of #95196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

996 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)(11,15)(12,14)(13,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,28,30)(29,32);(1,3)(2,5,6,7)(4,8)(10,14)(11,12)(15,16)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32);(2,6)(5,7)(9,11,13,12)(10,14,16,15)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gi,b^a=bd,c^a=ch,d^a=dg,f^a=fghi,c^b=cg,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240707, #72240809, #72257786, …
maximal normal subgroup of	#72240707, #72240809, #72257786, …
maximal quotient of	#72240714, #72240722

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2409);