#201398

order	512
nilpotency class	3
isoclinism	isoclinic class of #201394
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,f^b=fi,d^c=dh,e^c=ei,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310728, #72310729, #72310744, …
maximal normal subgroup of	#72310728, #72310729, #72310744, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108594);