#127994

order	512
nilpotency class	4
isoclinism	isoclinic class of #127645
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

439 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=hi,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=chi,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319637, #72319644, #72319660, …
maximal normal subgroup of	#72319637, #72319644, #72319660, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35190);