#348134

order	512
nilpotency class	3
isoclinism	isoclinic class of #348128
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#29684	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1037 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:

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

Polycyclic presentation:

2^9|b^2=ei,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268350, #72268353, #72268354, …
maximal normal subgroup of	#72268350, #72268353, #72268354, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255330);