#483421

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#106
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

480 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,e^2=g,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ch,d^b=di,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216672, #72216906, #72216937, …
maximal normal subgroup of	#72216672, #72216906, #72216937, …
maximal quotient of	#72285971, #72286101, #72286205, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390617);