#208058

order	512
nilpotency class	3
isoclinism	isoclinic class of #161894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=fhi,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^b=dh,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115254);