#7633646

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

5936 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)(3,7)(4,8)(5,6)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,22)(21,24);(2,6)(7,8)(17,19)(18,21)(20,24)(22,23);(9,11)(10,14)(12,15)(13,16)(18,22)(19,24);(10,15)(11,16)(18,22)(21,23);(1,3,5,4)(2,7,6,8);(1,4,5,3)(2,8,6,7)(9,12,13,14)(10,11,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72239530, #72239531, #72260547, …
Fitting subgroup of	#72239530, #72260486, #72261830
maximal subgroup of	#72233992, #72234084, #72238590, …
maximal normal subgroup of	#72233992, #72234084, #72238590, …
maximal quotient of	#72234052, #72260758, C₂×#7633646, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540842);