#182928

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18,21,25,22,26,33,39)(19,27,30,41,29,23,34,36)(20,24,31,37,32,38,44,47)(28,40,43,48,42,35,45,46);(1,3)(2,6,7,14)(4,11)(5,10)(8,9,12,15)(13,16)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,35,38,40)(25,36,39,41)(31,43)(32,42)(33,34)(37,46,47,48)(44,45);(3,10)(6,14)(9,15)(11,16)(17,20,21,31,22,32,33,44)(18,24,25,37,26,38,39,47)(19,28,30,43,29,42,34,45)(23,35,36,46,27,40,41,48);(2,7)(6,14)(8,12)(9,15)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,b^a=be,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239568, #72239730, #72239742, …
maximal normal subgroup of	#72239568, #72239730, #72239742, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90124);