#7851945

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

8220 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)(13,14)(15,18)(16,21)(17,25)(19,20)(22,27)(23,26)(24,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,19,16)(14,18,20,21)(17,23,27,24)(22,28,25,26);(5,7)(6,9)(8,12)(10,11)(14,20)(15,16)(22,25)(23,24);(13,16,19,15)(14,21,20,18)(17,24,27,23)(22,26,25,28);(5,7)(6,10)(8,12)(9,11)(13,17,19,27)(14,22,20,25)(15,23,16,24)(18,28,21,26);(5,6)(7,9)(8,11)(10,12)(13,18,19,21)(14,15,20,16)(17,26,27,28)(22,24,25,23)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279324, #72282583, #72282949, …
maximal normal subgroup of	#72279324, #72282583, #72282949, …
maximal quotient of	#72281106, #72281123, #72282249

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7759141);