#207903

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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309277, #72309287, #72309316
maximal normal subgroup of	#72309277, #72309287, #72309316

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115099);