#8932883

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 24 points:

(2,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18,21,23)(19,22,24,20);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,24)(18,22,23,20);(3,7)(6,8)(10,14)(13,15)(19,24)(20,22);(2,5)(6,8)(11,16)(13,15)(18,23)(19,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,20)(21,23)(22,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340917, #72341125
maximal subgroup of	#72227365, #72232440, #72233990, …
maximal normal subgroup of	#72227365, #72232440, #72233990, …
maximal quotient of	#72222747, #72232486, #72232605, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840079);