#7670725

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

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

11474 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283364, #72283392, #72283511, …
maximal normal subgroup of	#72283364, #72283392, #72283511, …
maximal quotient of	C₂×#7670725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577921);