#8932885

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

10340 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,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

derived subgroup of	#72227878, #72237983, C₂×#72227294, …
Fitting subgroup of	#72227766, #72227775, #72231801, …
maximal subgroup of	#72216396, #72216432, #72216464, …
maximal normal subgroup of	#72216396, #72216432, #72216464, …
maximal quotient of	#72216292, #72231761, C₂×#8932885, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840081);