#116000

order	512
nilpotency class	3
isoclinism	isoclinic class of #115995
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fh,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23196);