#6345759

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
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 ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1603 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287543, #72287566, #72287570, …
maximal normal subgroup of	#72287543, #72287566, #72287570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252955);