#6344955

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60559	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,f^b=fh,d^c=di,e^c=eh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322115
maximal normal subgroup of	#72322115
maximal quotient of	#72343892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252151);