#7641807

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(5,6,9,7)(8,11,12,10)(13,14,19,15)(16,22,25,20)(17,23,18,21)(24,27,26,28);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15,19,14)(16,22,25,20)(17,21,18,23)(24,27,26,28);(6,7)(10,11)(13,16)(14,20)(15,22)(17,24)(18,26)(19,25)(21,27)(23,28);(5,8,9,12)(6,10,7,11)(13,17,19,18)(14,21,15,23)(16,24,25,26)(20,27,22,28);(16,25)(20,22)(24,26)(27,28);(13,18,19,17)(14,21,15,23)(16,26,25,24)(20,27,22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279663, #72280047, #72280152, …
maximal normal subgroup of	#72279663, #72280047, #72280152, …
maximal quotient of	#72304129, C₂×#7641807

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549003);