#372908

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14103
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=fg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,e^b=ei,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319788
maximal normal subgroup of	#72319788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280104);