#7920205

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

7636 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)(5,6,9,11)(7,12,8,10)(13,14,18,22)(15,19,25,28)(16,20,24,27)(17,21,23,26);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,23)(17,24)(18,25)(20,26)(21,27)(22,28);(5,6)(7,12)(8,10)(9,11)(13,16)(14,20)(15,17)(18,24)(19,21)(22,27)(23,25)(26,28);(5,7,9,8)(6,10,11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28);(5,6)(7,10)(8,12)(9,11)(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,28)(22,27);(5,8,9,7)(6,10,11,12)(13,17,18,23)(14,21,22,26)(15,24,25,16)(19,27,28,20)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238144, #72238164, #72238245
maximal normal subgroup of	#72238144, #72238164, #72238245
maximal quotient of	#72263457

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7827401);