#190076

order	512
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

998 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302832, #72303078
maximal normal subgroup of	#72302832, #72303078

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97272);