#829128

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=fhi,d^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328279, #72328281, #72328301
maximal normal subgroup of	#72328279, #72328281, #72328301

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 736324);