#6345725

order	512
nilpotency class	3
isoclinism	isoclinic class of #62253
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	C₂×#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1588 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dhi,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=dhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343005, #72343014, #72343047
maximal normal subgroup of	#72343005, #72343014, #72343047
maximal quotient of	#72343332, #72343555, #72343800, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252921);