#7942031

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

8405 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)(5,6,9,10)(7,8,11,12)(13,14)(15,24)(16,27)(17,18)(19,23)(20,26)(21,28)(22,25);(1,2)(3,4)(13,15,19,26)(14,20,23,24)(16,25,28,17)(18,21,22,27);(6,10)(7,11)(13,16,19,28)(14,21,23,27)(15,17,26,25)(18,24,22,20);(5,7)(6,8)(9,11)(10,12)(13,16)(14,21)(15,25)(17,26)(18,24)(19,28)(20,22)(23,27);(5,8)(6,7)(9,12)(10,11)(13,17,19,25)(14,22,23,18)(15,16,26,28)(20,27,24,21);(13,18,19,22)(14,17,23,25)(15,21,26,27)(16,24,28,20)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,e^2=i,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	#72256676, #72256680, #72256685, …
maximal normal subgroup of	#72256676, #72256680, #72256685, …
maximal quotient of	C₂×#7942031

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7849227);