#7664541

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

6052 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227827, #72227831
Fitting subgroup of	#72227827, #72227829, #72227836
maximal subgroup of	#72227827, #72227828, #72227830, …
maximal normal subgroup of	#72227827, #72227828, #72227830, …
maximal quotient of	#72268012, #72302873

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7571737);