#8093716

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

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

7092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227577, #72227791
Fitting subgroup of	#72227785, #72227794
maximal subgroup of	#72213531, #72227412, #72227413, …
maximal normal subgroup of	#72213531, #72227412, #72227413, …
maximal quotient of	#72213467, #72240918, #72266737, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000912);