#7648641

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6584 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227964, #72227974, #72267909
Fitting subgroup of	#72227964, #72227971, #72227976
maximal subgroup of	#72227964, #72227967, #72227973, …
maximal normal subgroup of	#72227964, #72227967, #72227973, …
maximal quotient of	#72240918, #72267604, #72267609, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555837);