#7715995

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

7646 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)(13,14,19,16)(15,21,17,20)(18,24,26,22)(23,27,25,28);(1,2)(3,4)(13,15)(14,20)(16,21)(17,19)(18,23)(22,27)(24,28)(25,26);(6,7)(10,11)(14,16)(15,17)(22,24)(23,25);(5,7,9,6)(8,11,12,10)(13,16,19,14)(15,21,17,20)(18,24,26,22)(23,28,25,27);(5,8,9,12)(6,10,7,11)(13,17,19,15)(14,21,16,20)(18,25,26,23)(22,28,24,27);(13,18)(14,22)(15,23)(16,24)(17,25)(19,26)(20,27)(21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238163, #72238210, #72238230
maximal normal subgroup of	#72238163, #72238210, #72238230
maximal quotient of	#72263522

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7623191);