#7915597

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

8698 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)(7,8)(9,10)(11,12)(13,14,19,21)(15,20,24,27)(16,26,22,17)(18,23,25,28);(1,2)(3,4)(13,15)(14,20)(16,25)(17,23)(18,22)(19,24)(21,27)(26,28);(6,10)(7,11)(14,21)(15,24)(16,22)(23,28);(5,7,9,11)(6,8,10,12)(13,16,19,22)(14,17,21,26)(15,25,24,18)(20,23,27,28);(5,8)(6,11)(7,10)(9,12)(13,17)(14,22)(15,23)(16,21)(18,20)(19,26)(24,28)(25,27);(13,18,19,25)(14,23,21,28)(15,22,24,16)(17,27,26,20)

Polycyclic presentation:

2^9|a^2=i,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	#72238142, #72238148, #72238258
maximal normal subgroup of	#72238142, #72238148, #72238258

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7822793);