#7630606

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

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

23658 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:

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

Polycyclic presentation:

2^9|f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72241294, #72241296
maximal subgroup of	#72240149, #72241292, #72241294, …
maximal normal subgroup of	#72240149, #72241292, #72241294, …
maximal quotient of	#72240154, #72240168, #72265992, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537802);