#246997

order	512
nilpotency class	3
isoclinism	isoclinic class of #246987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#51903	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278562, #72278907, #72285747
maximal normal subgroup of	#72278562, #72278907, #72285747

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154193);