#7744957

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

8202 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)(13,14,18,21)(15,17,22,19)(16,20,26,28)(23,25,24,27);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(15,22)(17,19)(23,24)(25,27);(6,9)(10,11)(13,15)(14,19)(16,24)(17,21)(18,22)(20,25)(23,26)(27,28);(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24);(5,7)(6,10)(8,12)(9,11)(13,17,18,19)(14,15,21,22)(16,25,26,27)(20,23,28,24);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302957, #72303001, #72303013
maximal normal subgroup of	#72302957, #72303001, #72303013

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7652153);