#7837474

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

7529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238144, #72238194, #72238223, …
maximal normal subgroup of	#72238144, #72238194, #72238223, …
maximal quotient of	#72263358

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7744670);